Extension of Paley Construction for Hadamard Matrix Shipra Kumari∗ and Hrishikesh Mahato† Department of Mathematics, Central University of Jharkhand, Ranchi-835205, India Abstract We have extended the Paley constructions for Hadamard matrices and obtained some se- ries of Hadamard matrices. Especially Paley construction-II is applicable for odd prime power q ≡ 1(mod 4) however our method is applicable for any odd prime power. Some of which are non-isomorphic to the matrices obtained by standard Paley constructions, Sylvester construc- tion, Williamson construction or combination of either of these constructions with Sylvester construction. In fact we have used the conference matrix of order prime power q in some differ- ent manner which is applicable for any odd prime power q or product of twin primes. Keywords: Conference matrix, Quadratic Residue, Extended Quadratic Character, Kronecker Product. AMS Subject classification: 05B20 1 Introduction: A Hadamard Matrix is a square matrix Hn of order n with entries from {−1, 1} which satisfies T T the condition HnHn = nIn, where In is the Identity matrix of order n and Hn is the transpose of Hn. The order of Hadamard Matrix is necessarily 1 , 2 or is multiple of 4. The converse of this statement seems to be true and is known as Hadamard Conjecture [4]. Historically, Hadamard matrices were first introduced by James Joseph Sylvester in 1867 [2]. arXiv:1912.10755v1 [math.CO] 23 Dec 2019 Sylvester presented a construction by considering that if H is a Hadamard matrix of order n then H H H −H is also a Hadamard matrix of order 2n. Starting with H = 1, we get a Hadamard matrix of order 2n for all non-negative integer n 1 1 H2n−1 H2n−1 H1 = 1 , H2 = , H2n = 1 −1 H n− −H n− 2 1 2 1 ∗E-mail: [email protected] †Corresponding author: E-mail: [email protected] 1 In this manner Sylvester constructed Hadamard matrix of order 2n for every positive integer n. Another popular construction of Hadamard matrix is Paley construction which was described in 1933 by the english mathematician Raymond Paley [11]. The Paley construction uses quadratic residues of a finite field GF (q), where q is an odd prime power. Two methods of Paley constructions has been distinguished by the types of q with q ≡ 1(mod 4) and q ≡ 3(mod 4). The properties of conference matrix are used in both the constructions. In Paley construction type-I for each q ≡ 3 (mod 4) there exist a Hadamard matrix of order q + 1 and in that of type-II for each q ≡ 1 (mod 4) there exist a Hadamard matrix of order 2(q + 1). In this article we use the conference matrices Q of order q = pr ( p is an odd prime and r ∈ Z+) in different manner and construct Hadamard matrices of order n(q + 1) for q ≡ 3(mod 4) and that of order 2k+1(q + 1) for q ≡ 1(mod 4), where n is the order of a known Hadamard matrix and k is any non-negative integer. It has been observed that some of Hadamard matrices of order 16, 40, 56, · · · cannot be constructed by Paley construction-I and construction-II directly. Our aim is to extend the Paley construction-I and II so that we can construct Hadamard matrices of above mentioned orders directly. Also we will discuss about existence of Hadamard matrices using twin primes and unifiy in same construction. 2 Preliminaries Definition 2.1. [6]Extended Quadratic character r Let q = p , where p is an odd prime, r is any positive integer and GF (q)= {0= α0, α1, α2, ..., αq−1}. Then the Extended quadratic character is a map χ defined on GF (q) as 1 ; if αi is quadratic residue in GF (q); χ(αi)= 0 ; if αi = 0 ; −1 ; if αi is quadratic nonresidue in GF (q). Definition 2.2. [6]Quadratic Residue An element a of GF (q) is said to be quadratic residue if it is a perfect square in GF (q) otherwise a is a quadratic non-residue. Definition 2.3. [4]Kronecker Product Let A = [aij] and B = [brs] be two square matrices of order m and n respectively. Then the Kro- necker Product (Direct Product ) of A and B is a square matrix of order mn and is given by a11B a12B . a1mB a21B a22B . a2mB . A ⊗ B = . . a B a B . a B m1 m2 mm Definition 2.4. [8] Sign changes of a sequence n The number of sign changes of a finite sequence {ak}k=1, ak = ±1 is the number obtained by counting the number of times +1 is followed by −1 and that of times −1 is followed by +1. 2 Definition 2.5. [8]Sign spectrum of a Hadamard matrix The row (column) sign spectrum of an n × n Hadamard matrix H is the sequence of numbers of sign changes appear in its rows (columns). Definition 2.6. Conference Matrix A Conference Matrix is a square matrix of order q with diagonal entries 0 and off-diagonal entries T ±1 which satisfies the condition QQ = qIq − Jq. Lemma 2.1. [6] Let F be a field then b χ(b)χ(b + c)= −1 if c 6= 0, where b, c ∈ F P Lemma 2.2. [6] The matrix Q = [qij] of order q × q where qij = χ(βj − βi), i, j = 0, 1, 2, ..., q − 1 has the following properties (a) Q is symmetric if q ≡ 1 (mod 4) and skew symmetric if q ≡ 3 (mod 4) T (b) QQ = qIq − J 3 Results: Consider Hn = [hij] is a Hadamard matrix of order n. Let us define a rectangular matrix E = [eij] of order nq × n whose elements are defined as follows h1j if 1 ≤ i ≤ q; h2j if q < i ≤ 2q . eij = (1) . h if n q < i nq nj ( − 1) ≤ where j= 1, 2, 3, ...n Infact, matrix E = [eij] is formed by repeating each row of Hadamard matrix Hn q times succes- st th th sively. i.e. the first q rows of E are repetitions of 1 row of Hn, (q + 1) row to 2q rows of E are nd repetitions of 2 row of Hn and so on. Similarly the matrix E′ of order n × nq is formed by repeating each column of Hadamard matrix Hn q times successively. Since rows and columns of Hadamard matrix is mutually orthogonal. So nq 0 · · · 0 0 nq · · · 0 ′ ′T E E = . = nqIn (2) . .. 0 · · · nq n×n 3 and, n · · · n 0 · · · 0 q | {z0 } n · · · n 0 . T q . HnE = . (3) . | {z } . . 0 n · · · n q n×nq | {z } Similarly T T T EHn = (HnE ) (4) and nJq 0 · · · 0 0 nJ · · · 0 T q EE = . = nJq ⊗ In (5) . .. 0 · · · nJ q nq×nq where Jq is a square matrix of order q with all entries 1. Theorem 3.1. Let q = pr, where p is an odd prime and r is any positive integer, such that q ≡ 3 (mod 4), then there exist a Hadamard matrix of order nq + n, where n is the order of a known Hadamard Matrix. Proof. Consider a square matrix A = [Q + I] ⊗ Hn of order nq, where Q is a conference matrix of order q. Then we see that −H E′ K = n E A forms a Hadamard matrix of order nq + n. We have T −H E′ −H E′ KKT = n n E A E A −H ET −HT ET = n n E A E′T AT T ′ ′T T ′ T HnHn + E E −HnE + E A = T ′ T T (6) −EHn + AE T EE + AA 4 Now T T AA = ([Q + I] ⊗ Hn) ([Q + I] ⊗ Hn) T T = ([Q + I] ⊗ Hn) ([Q + I] ⊗ Hn ) T T T = ([QQ + Q + Q + I]) ⊗ HnHn T T = ([QQ + Q + Q + I]) ⊗ nIn T T = ([QQ + I]) ⊗ nIn As, Q = −Q for q ≡ 3(mod 4) = ([qIq − Jq + Iq]) ⊗ nIn = ((q + 1)Iq − Jq) ⊗ nIn = n(q + 1)Iq ⊗ In − nJq ⊗ In As T EE = nJq ⊗ In So, T T AA + EE = n(q + 1)Iq ⊗ In − nJq ⊗ In + nJq ⊗ In T T ⇒ AA + EE = n(q + 1)Iq ⊗ In (7) As the number of quadratic residues and that of quadratic non-residues in GF (q) are same which q−1 ′ ′ is 2 by the definition of Q the number of +1 s and −1 s in each row and column of matrix Q + I q−1 q−1 are 2 +1 and 2 respectively. Therefore n · · · n q | {z } n · · · n q ′ T | {z } E A = (8) . .. n · · · n q n×nq | {z } Using equation (3) and equation (8) ′ T T E A − HnE = 0 (9) Taking transpose both sides ′T T AE − EHn = 0 (10) From the equation (2) and Hadamard matrix Hn we have T ′ ′T HnHn + E E = nIn + nqIn = (nq + n)In (11) 5 Combining equation (7), (9), (10) , (11) equation (6) becomes, nq + n 0 · · · 0 nq n T 0 + · · · 0 KK = . = (nq + n)Inq+n . .. 0 · · · nq + n Therefore K is a Hadamard matrix of order nq + n. Remark: In particular this method coincides with Paley construction-I for n = 1 Theorem 3.2. Let q = pr, where p is an odd prime and r is any positive integer, such that q ≡ 1 (mod 4), then there exist a Hadamard matrix of order 2k+1(q + 1), where k is any non- negative integer. Proof. Consider a square matrix A = [Qq ⊗±(RH2R)+ Iq ⊗ H2] ⊗ H2k (12) where H2k = H2 ⊗ H2 ⊗···⊗ H2 and Q is a conference matrix of order q.